Langmuir 2004, 20, 5185-5188
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Phase Diagrams of Nonionic Foam Films: New Interpretation of Disjoining Pressure vs Thickness Curves Cosima Stubenrauch* and Reinhard Strey Institut fu¨ r Physikalische Chemie, Universita¨ t zu Ko¨ ln, Luxemburger Strasse 116, D-50939 Ko¨ ln, Germany Received February 16, 2004. In Final Form: May 6, 2004 Recently we constructed phase diagrams for thin foam films stabilized by a nonionic surfactant. The idea was born by synopsis of various disjoining pressure (Π) versus thickness (h) curves of foam films resembling p-Vm isotherms of real gases. The new concept of interpreting the Π-h curves of foam films in terms of phase diagrams allows us to describe experimental observations much more precisely. Three logical consequences will be discussed here to illustrate the strength of this approach. First, the observation is explained that common black films (CBF) rupture or form a Newton black film (NBF) within a certain pressure range rather than at a defined pressure. Both observations can be rationalized by invoking a nucleation process of holes or of the thinner NBF, respectively, in close analogy to the vapor to liquid condensation. Second, the question whether the CBF to NBF transition is discrete or continuous is answered by analyzing under which conditions the supercritical state of a foam film can be reached. Third, the evidence of corresponding states is discussed.
1. Introduction The thickness h of foam films can be measured as a function of the disjoining pressure Π. The resulting Π-h curves of foam films stabilized with nonionic surfactants measured at various concentrations1-10 resemble the p-Vm isotherms of a real gas measured at various temperatures (p is the pressure and Vm is the molar volume of the gas).11 The striking similarity led us to apply the van der Waals approach for real gases to thin foam films. We found that if the thickness h takes the role of Vm and the disjoining pressure Π replaces the ordinary pressure p, all features of the p-Vm phase diagram are found in the resulting Π-h diagram.12 As is seen in Figure 1, our analysis resulted in a Π-h phase diagram for a “model thin foam film” by analogy with the p-Vm phase diagram of a real gas including a two-phase region, a critical point, and spinodals. In this model the thicker, electrostatically stabilized common black film (CBF) corresponds to the gas phase, while the compact Newton black film (NBF) corresponds to the dense liquid. It was shown in ref 12 that appropriate tuning parameters can be identified in order to justify the comparison between a real gas and a foam film. While it is the temperature T for the real gas which determines the phase * Corresponding author. (1) Kolarov, T.; Cohen, R.; Exerowa, D. Colloids Surf. 1989, 42, 49. (2) Black, I. J. Thesis, Reading, 1992. (3) Bergeron, V.; Waltermo, A.; Claesson, P. M. Langmuir 1996, 12, 1336. (4) Exerowa, D.; Kruglyakov P. M. Foam and Foam films-Theory, Experiment, Application; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1998. (5) Karraker, K. A. Thesis, Berkeley, 1999. (6) Persson, C. M.; Claesson, P. M.; Johansson, I. Langmuir 2000, 16, 10227. (7) Karraker, K. A.; Radke, C. J. Adv. Colloid Interface Sci. 2002, 96, 231. (8) Stubenrauch, C.; Schlarmann, J.; Strey, R. Phys. Chem. Chem. Phys. 2002, 4, 4504. Stubenrauch, C.; Schlarmann, J.; Strey, R. Phys. Chem. Chem. Phys. 2003, 5, 2736. (9) Schlarmann, J.; Stubenrauch, C. Tenside, Surfactants, Deterg. 2003, 40, 190. (10) Schlarmann, J.; Stubenrauch, C.; Strey, R. Phys. Chem. Chem. Phys. 2003, 5, 184. (11) For a modern treatise of physical chemistry see e.g.: Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; Oxford University Press: Oxford, 2002. (12) Stubenrauch, C.; Kashchiev, D.; Strey, R. JCIS, in press.
Figure 1. (top, a) Schematic Π-h phase diagram of a nonionic foam film. At large thicknesses and low disjoining pressures a common black film (CBF) is formed, whereas at low thicknesses the Newton black film (NBF) is the stable phase. These two phases are separated by a two-phase region where a CBF and a NBF coexist. (bottom, b) Π-h phase diagram constructed from calculated Π-h curves using actual experimental parameters. The coordinates of the critical point are q0,c ) 11 mC m-2, Πc ) 0.89 × 105 Pa, and hc ) 5 nm. Details are given in ref 12.
behavior of the system, it is the surface charge density q0 in the case of the thin foam film. Thus, we proposed to
10.1021/la0495940 CCC: $27.50 © 2004 American Chemical Society Published on Web 05/22/2004
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Figure 2. Enlarged part of the Π-h phase diagram presented in Figure 1b. Note that compared to Figure 1b the scales of both the y- and the x-axes are different. In addition, three experimental Π-h curves of the nonionic sugar surfactant β-C12G2 and the respective calculated curves are given. The experimental data are taken from ref 8.
speak of Π-h curves rather than of Π-h isotherms because they are measured at both constant T and constant q0. Three schematic Π-h curves are drawn in Figure 1a illustrating different situations that may be expected. In Figure 1b we calculated the actual shape of the two-phase region and the critical point. The somewhat awkward shape is a consequence of the extremely short range of the hard wall interaction, which makes the binodal and spinodal collapse into the (almost) vertical line. In the present paper we interpret experimental Π-h curves of films stabilized by the nonionic sugar surfactant n-dodecyl-β-D-glucoside (β-C12G2) in terms of phase diagrams. The stability of a CBF as well as the transition from a CBF to a NBF will be discussed. Our new approach to thin foam films can also be used to predict whether the transition from a CBF to a NBF takes place via a stepwise or a continuous thinning. 2. Phase Transitions of a CBF Applying an external capillary pressure to a CBF, one observes either a film rupture or a transition to a NBF. Repeating the experiments one observes the respective phase transition usually not to occur at a defined pressure, but within a certain pressure range. With respect to the CBFs of the nonionic sugar surfactant n-octyl-β-D-glucoside (β-C8G1) Bergeron et al.3 remarked: “the films rupture in a somewhat statistical fashion”. The same holds true for the CBF-NBF transition.8 A promising concept is to treat the film surfaces as fluctuating surfaces instead of solid non-deforming walls as is done in the DLVO theory. The correlation between fluctuations and DLVO forces has been treated only qualitatively, though.13,14 Both experimental and theoretical work is lacking to clarify and quantify this point. In the present paper we propose an explanation from a purely thermodynamic point of view based on the notation used in fundamental phase diagrams. With this new approach the “somewhat statistical fashion” can be explained and shown to be in fact stochastic in nature, i.e., a nucleation phenomenon. In Figure 2, a part of the phase diagram already presented in Figure 1b is enlarged. The construction of (13) Bergeron, V. Langmuir 1997, 13, 3474. (14) Bergeron, V. J. Phys.: Condens. Matter 1999, 11, R215.
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this phase diagram was based on experimental Π-h curves measured for foam films which were stabilized by aqueous solutions of the nonionic sugar surfactant β-C12G2. Three of these Π-h curves as well as the respective calculated curves are shown in Figure 2. The measurements were performed as a function of the surfactant concentration at a constant electrolyte concentration of c(NaCl) ) 10-4 M. In the following, the stability of a CBF stabilized by β-C12G2 as well as its transition to a NBF will be discussed. For the sake of clarity, we will first focus on the rupture of the CBF. A film stabilized by β-C12G2 at a solution concentration of c ) 6.8 × 10-6 M ruptures at some high pressure. A series of repeated, independent measurements led to reproducible data points lying on the same Π-h curve. However, the pressure at which the CBF ruptured varied between 4000 and 6000 Pa. Inspection of Figure 2 reveals that this pressure range is located in the metastable region of the phase diagram between the binodal and the spinodal. In this region the phase transition apparently takes place via a nucleation process, which, in turn, is a stochastic event. Hence, knowing the phase diagram, we can predict the pressure range in which the CBF ruptures and expect the rupture to be stochastic as long as the applied pressure lies in the metastable region. The same observation, namely, a pressure range in which the film ruptured, was made a the solution with c ) 6.8 × 10-5 M, i.e., 1 order of magnitude higher surfactant concentration. The interesting difference is that during some of the experiments a transition to a NBF was observed instead of a film rupture. The film was not very stable, though. Increasing the concentration further to 1.4 × 10-4 M led to the formation of a stable NBF at pressures ranging from 1000 to 2000 Pa. Thus, if the CBF is no longer able to resist the applied pressure, i.e., if the applied pressure exceeds a certain maximum disjoining pressure Πmax for the CBF, it either ruptures or forms a NBF, depending on the surfactant concentration.15 In the metastable region of the phase diagram the phase transition takes place via a nucleation process. In the first case the nuclei obviously lead to holes, whereas in the second they are nuclei of the NBF. A characteristic feature of a nucleation-mediated process is an increasing probability of nucleation with increasing depth of entering the metastable region. This leads to the experimental observation that the transitions take place in a somewhat irregular fashion and thus occur within a certain pressure range rather than at a defined pressure. A closer look at Figure 2 reveals a deviation at low film thicknesses between the experimental data and the respective calculated curves. Although the experimental data are adequately described by the DLVO theory down to thicknesses of around 10-15 nm for all concentrations, an up-turn of the experimental Π-h curves is observed at smaller thicknesses. Possible reasons for this up-turn are discussed in ref 16. Presently, a satisfactory quantitative explanation of the up-turn cannot be given. It appears as if the classical DLVO theory cannot describe accurately the experimental data over the whole thickness range. In Figure 2, one sees that according to the DLVO calculations it is impossible to form a stable film at a pressure of 6000 Pa and a surfactant concentration of c ) 6.8 × 10-6 M. Theoretically this particular film is expected to rupture (15) Not only the surfactant concentration but also the structure of the surfactant dictates the stability of both the CBFs and the NBFs, which is discussed in detail in ref 9. First of all, a minimum length of the alkyl chain is required to stabilize thin liquid films in general. If this condition is fulfilled it is the size of the headgroup which determines whether a stable NBF can be formed. (16) Stubenrauch, C.; Rojas, O. J.; Schlarmann, J.; Claesson, P. M. Langmuir, in press.
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spontaneously because approaching a spinodal nucleation rates become very high. The experimental fact that it does not rupture shows the need for an improved theoretical description, i.e., an improved “equation of state” for the foam film. Although a more accurate equation of state will slightly shift the spinodal, we expect the general features of the phase diagram discussed not to change. In conclusion, the electrostatic double layer forces stabilizing a CBF create an energy barrier, i.e., a maximum disjoining pressure Πmax. In the metastable region of the phase diagram where the local disjoining pressure is smaller than the maximum value, i.e., Π < Πmax, the transition to a NBF (and the subsequent ruptures) is governed by a nucleation process and thus so are stochastic events. This is in perfect agreement with experimental observations made so far3,8,17,18 and in the early work of De Vries, who discussed nucleated rupture of thick foam films as early as 1958.19 Whenever the applied pressure exceeded Πmax, the respective phase transition was found to take place spontaneously.18 In other words, the phase transition then happens because the nucleation rate becomes very high before a spinodal decomposition can occur in accordance with the construction of the phase diagram described in ref 12 and the original work of Vrij.20,21 3. Discrete and Continuous CBF-NBF Transitions The strength of our new approach to foam films is that we can explain whether the transition from a CBF to a NBF film is discrete or continuous. While for most surfactants investigated so far a stepwise transition from the thick CBF to the thin NBF has been observed, for some systems a continuous transition has been found. The first continuous CBF-NBF transition was observed for aqueous solutions of eicosaoxyethylene nonylphenol ether (NP20) by Kolarov et al.1 Ten years later it was shown in an extensive disjoining pressure study5 that the CBF-NBF transition is discrete for C10E4 and continuous for C10E8, because in the latter case the film is stabilized by short-range repulsive forces before attractive van der Waals forces dominate the interactions. In other words, a crossover from a discrete to a continuous NBF formation is expected between E4 and E8. Experiments to check this expectation are underway. However, before the more timeconsuming measurements are performed, some predictions can be made with simple calculations applying our new approach. The question to be answered is if it is possible to tune the parameters such that a continuous CBF-NBF transition results from the phase diagram. The prospect is already given in Figure 1a, which illustrates that a continuous CBF-NBF transition takes place in the supercritical region, i.e., at conditions where a distinction between a CBF and a NBF is no longer possible. We will call the respective film a supercritical film. According to Figure 1b, for a NBF of 5 nm thickness such a state is reached at Π > Πc ) 0.89 × 105 Pa, which corresponds to a surface charge density of q0 > q0,c ) 11.0 mC m-2. Is it possible to meet these conditions experimentally? To answer this question we have to remember that the surface (17) Exerowa, D.; Kolarov, T.; Khristov, Khr. Colloids Surf. 1987, 22, 171. (18) Casteletto, V.; Cantat, I.; Sarker, D.; Bausch, R.; Bonn, D.; Meunier, J. Phys. Rev. Lett. 2003, 90, 048302. (19) De Vries, A. J. Recl. Trav. Chim. Pays-Bas 1958, 77, 383 and 441. (20) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23. (21) Vrij, A.; Overbeek, J. Th. G. J. Am. Chem. Soc. 1968, 90, 3074.
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charge density q0 of the water/air surface has an upper limit of 1.0-1.7 mC m-2 in the presence of nonionic surfactants3,4,8,16,22 for the chosen electrolyte concentration of 10-4 M. Consequently, the supercritical state of an aqueous thin foam film cannot be reached experimentally if the corresponding critical surface charge density is higher than 1.7 mC m-2, as is the case for the present “model thin foam film”. In other words, conditions are required under which critical surface charge densities below 1.7 mC m-2 can be reached. Considering for a moment the van der Waals equation of state in the form12
p)
a RTb RT + Vm(Vm - b) Vm2 Vm
(1)
with R being the gas constant one recognizes that the competition between the van der Waals attraction (second term) and the pressure of the ideal gas (third term) are crucial for the nonideal behavior to occur. The divergence of the (first) hard core term becomes dominating as Vm approaches the order of b. At high T ideal behavior is observed. Lowering T, at some T ) Tc the nonideality expresses itself in the occurrence of the two-phase region. For the van der Waals equation pc ) a/27b2 and Tc ) 8a/27Rb. Thus, for a given attraction parameter a, pc can be lowered by increasing the steric repulsion parameter b which also lowers Tc. Let us now consider the equation of state for the foam films
Π(h) ) Πsr(h) + ΠvdW(h) + Πelec(h)
(2)
with
[( ) ( ) ]
Πsr(h) = kT Γ3/2
2hhead hcore
ΠvdW(h) ) -
Πelec(h) )
(
9/4
-
hcore 2hhead
A 6πh3
2πq02 2 sech(κh/2) 1 + tanh(κh/2)
3/4
(3)
(4)
)
2
(5)
where k is the Boltzmann constant, Γ is the surface concentration of the surfactant, hhead is the length of the headgroup, hcore is the thickness of the film core consisting of the headgroups and water, A is the Hamaker constant, c is the electrolyte concentration, and κ is the inverse Debye length. Note that the total film thickness equals h ) hcore + 2htail where the length of the hydrophobic tail htail is considered to be constant. More details can be found in ref 12. Looking at eq 1, eq 2, and eq 5, one sees that q0 plays here the role that T plays in the p-V case. Therefore, increasing the headsize of the surfactant hhead, i.e., the steric repulsion, at a given attractive van der Waals attraction will necessarily lead to a lowering of q0. Accordingly, the critical point will shift toward lower Πc and hence to lower q0,c values if the thickness of the NBF is made to increase. The results of the respective calculations are presented in Figure 3, where the dependence of the critical surface charge density q0,c is shown as a function of the NBF thickness hNBF. Referring to a surface charge density of 1.7 mC m-2 as the upper limit, we can extract from Figure 3 that the (22) Exerowa, D.; Zacharieva, M.; Cohen, R.; Platikanov, D. Colloid Polym. Sci. 1979, 257, 1089.
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From the above discussion it is obvious that the shapes of the Π-h curves and transitions of thin foam films show a striking similarity with p-Vm curves and the vaporliquid transitions. As van der Waals anticipated in 1873 and realized in 1881 for the p-Vm case the different systems should scale and collapse if plotted in the appropriate reduced variables. Such a unified treatment was a milestone for the understanding of real gas systems. We hope (actually are confident) that our analysis of thin films can proceed along the same lines and follow his example. 4. Conclusions
Figure 3. Critical surface charge densities q0,c as a function of the NBF thickness hNBF. Calculations are based on the assumption that the NBF is incompressible. Note that a slight pressure dependence of the thickness is already seen for the NBF of C12E6 which is around 6 nm.9,16 Thus, the assumption of an incompressible NBF is only a rough approximation for NBFs thicker than 6 nm.
supercritical state can be reached experimentally for systems whose NBFs have a thickness of h g 11 nm. This is in absolute agreement with the results obtained for other nonionic surfactants, namely, NP20,1 amphiphilic diblock23 as well as triblock copolymers,24-26 and for C10E8.5 In all cases, the film thicknesses do not fall below 10 nm and continuous thinning with increasing pressure is observed. Note that the critical parameters and thus the thickness above which a continuous transition is expected strongly depend on the electrolyte concentration. It has to be kept in mind that all calculations in the present work were performed at an electrolyte concentration of 10-4 M. In most of the publications quoted above higher electrolyte concentrations were chosen so that care has to be taken when comparing the data quantitatively. Moreover, except in ref 5, the thicknesses of the water core are given, whereas in the present work the thickness h includes the adsorbed surfactant layers. (23) Claesson, P. M.; Kjellin U. R. M. In Modern characterization methods of surfactant systems; Binks, B. P., Ed.; Marcel Dekker: New York, 1999. (24) Sedev, R.; Ivanova, R.; Kolarov, T.; Exerowa, D. J. Dispersion Sci. Technol. 1997, 18, 751. (25) (a) Sedev, R.; Ne´meth, Zs.; Ivanova, R.; Exerowa, D. Colloids Surf., A 1999, 149, 141. (b) Sedev, R.; Exerowa, D. Adv. Colloid Interface Sci. 1999, 83, 111. (26) Sedev, R.; Exerowa, D.; Findenegg, G. H. Colloid Polym. Sci. 2000, 278, 119.
The observation that both the rupture of the CBF as well as the transition from the CBF to the NBF takes place within a certain pressure range rather than at a well-defined pressure can be explained by regarding the Π-h curves as parts of a phase diagram. In doing so it becomes obvious that these two processes are nucleationmediated and thus stochastic events. Our new approach to thin foam films can also be used to predict whether the transition from a CBF to a NBF takes place via a stepwise or a continuous thinning. It was shown under which conditions the supercritical state of a foam film can be reached, which means that the disjoining pressure Π changes continuously with h and a distinction between CBF and NBF is no longer possible. We believe that the interpretation of foam film properties in terms of fundamental phase diagrams is a powerful tool because they are independent of the detailed nature of the surfactant. As in other cases the theoretical treatment lends itself to a scaling description in terms of corresponding states. The constructed phase diagram seen in Figure 1b is valid for all nonionic surfactants, the only conditions to be fulfilled being that the NBF has a thickness of 5 nm and that the electrolyte concentration equals 10-4 M. Changing this concentration leads to changes in both the Debye length and the surface charge density, which, in turn, changes the position of the phase boundaries. However, the general features are not changed. What is still missing is the application of the present concept to foam films stabilized by ionic surfactants. The main problem in doing so is the fact that the ionic strength changes when the surfactant concentration is changed. Thus, at each concentration, a different electrolyte content is needed to adjust the ionic strength. Work with respect to the construction of a phase diagram for ionic surfactants is under way. Acknowledgment. C.S. is indebted to the DFG and the Fond der Chemischen Industrie for financial support. Illuminating discussions with D. Kashchiev are gratefully acknowledged. LA0495940
